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Concept# Forward kinematics

Summary

In robot kinematics, forward kinematics refers to the use of the kinematic equations of a robot to compute the position of the end-effector from specified values for the joint parameters.
The kinematics equations of the robot are used in robotics, computer games, and animation. The reverse process, that computes the joint parameters that achieve a specified position of the end-effector, is known as inverse kinematics.
The kinematics equations for the series chain of a robot are obtained using a rigid transformation [Z] to characterize the relative movement allowed at each joint and separate rigid transformation [X] to define the dimensions of each link. The result is a sequence of rigid transformations alternating joint and link transformations from the base of the chain to its end link, which is equated to the specified position for the end link,
where [T] is the transformation locating the end-link. These equations are called the kinematics equations of the serial chain.
In 1955, Jacques Denavit and Richard Hartenberg introduced a convention for the definition of the joint matrices [Z] and link matrices [X] to standardize the coordinate frame for spatial linkages. This convention positions the joint frame so that it consists of a screw displacement along the Z-axis
and it positions the link frame so it consists of a screw displacement along the X-axis,
Using this notation, each transformation-link goes along a serial chain robot, and can be described by the coordinate transformation,
where θi, di, αi,i+1 and ai,i+1 are known as the Denavit-Hartenberg parameters.
The kinematics equations of a serial chain of n links, with joint parameters θi are given by
where is the transformation matrix from the frame of link to link . In robotics, these are conventionally described by Denavit–Hartenberg parameters.
The matrices associated with these operations are:
Similarly,
The use of the Denavit-Hartenberg convention yields the link transformation matrix, [i-1Ti] as
known as the Denavit-Hartenberg matrix.

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In mechanical engineering, a kinematic chain is an assembly of rigid bodies connected by joints to provide constrained motion that is the mathematical model for a mechanical system. As the word chain suggests, the rigid bodies, or links, are constrained by their connections to other links. An example is the simple open chain formed by links connected in series, like the usual chain, which is the kinematic model for a typical robot manipulator. Mathematical models of the connections, or joints, between two links are termed kinematic pairs.

In computer animation and robotics, inverse kinematics is the mathematical process of calculating the variable joint parameters needed to place the end of a kinematic chain, such as a robot manipulator or animation character's skeleton, in a given position and orientation relative to the start of the chain. Given joint parameters, the position and orientation of the chain's end, e.g. the hand of the character or robot, can typically be calculated directly using multiple applications of trigonometric formulas, a process known as forward kinematics.

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